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+ | {{Types of Primes Infobox |
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− | {{Types of Primes Infobox|title=Balanced Primes|discov=N/A|amount=Conjectured infinite|expression=A prime which is the mean of the previous prime and the following prime|1st=[[5]], [[53]], [[157]], [[173]], [[211]], [[257]], [[263]]}} |
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+ | |title=Balanced Primes |
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+ | |discov=N/A |
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+ | |amount=Conjectured infinite |
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+ | |expression=A prime which is the mean of the previous prime and the following prime |
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+ | |1st=[[5]], [[53]], [[157]], [[173]], [[211]], [[257]], [[263]]}} |
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'''{{PAGENAME}}''' are primes which are the mean of the next primes number and the previous prime number. This can be expressed as <math>p_n = {{p_{n - 1} + p_{n + 1}} \over 2}.</math>, where <math>{{p_n}}</math> is the nth prime number and <math>{{p_{n - 1}}}</math> and <math>{{p_{n + 1}}}</math> are the previous prime number and the next prime number, respectively. 5 is the first balanced prime, as 3+7/2 =5. It is thought, although not proven, that there are infinitely many balanced primes. In theory, in order to be a balanced prime, the difference between the nth prime number and the n+1th prime number can be any even number, as long as it is the same as the gap between the nth and the n-1th. Most of said even numbers are multiples of three, as a result of the pigeonhole principle. If the difference is not a multiple of three, one of the three numbers n-x, n or n+x will be a multiple of three. |
'''{{PAGENAME}}''' are primes which are the mean of the next primes number and the previous prime number. This can be expressed as <math>p_n = {{p_{n - 1} + p_{n + 1}} \over 2}.</math>, where <math>{{p_n}}</math> is the nth prime number and <math>{{p_{n - 1}}}</math> and <math>{{p_{n + 1}}}</math> are the previous prime number and the next prime number, respectively. 5 is the first balanced prime, as 3+7/2 =5. It is thought, although not proven, that there are infinitely many balanced primes. In theory, in order to be a balanced prime, the difference between the nth prime number and the n+1th prime number can be any even number, as long as it is the same as the gap between the nth and the n-1th. Most of said even numbers are multiples of three, as a result of the pigeonhole principle. If the difference is not a multiple of three, one of the three numbers n-x, n or n+x will be a multiple of three. |
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+ | ==Examples== |
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+ | *[[53]] is a balanced prime. The previous prime number is [[47]], while the next prime number is [[59]], both of which are 6 numbers away from 53. |
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+ | *[[5]] is a balanced prime. The pervious prime number is [[3]], while the next prime number is [[7]], both of which are 2 numbers away from 5. |
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==The First Few Balanced Primes== |
==The First Few Balanced Primes== |
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− | <gallery widths="50"> |
+ | <gallery widths="50" navigation="true"> |
5.png|link=5 |
5.png|link=5 |
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</gallery> |
</gallery> |
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− | [[Category:Groups of primes]] |
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[[Category:Types of Primes]] |
[[Category:Types of Primes]] |
Revision as of 03:18, 23 January 2017
Balanced Primes
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Basic Info
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Discovered by | N/A |
Number of | Conjectured infinite |
Description | A prime which is the mean of the previous prime and the following prime |
First Few | 5, 53, 157, 173, 211, 257, 263 |
Balanced Primes are primes which are the mean of the next primes number and the previous prime number. This can be expressed as , where is the nth prime number and and are the previous prime number and the next prime number, respectively. 5 is the first balanced prime, as 3+7/2 =5. It is thought, although not proven, that there are infinitely many balanced primes. In theory, in order to be a balanced prime, the difference between the nth prime number and the n+1th prime number can be any even number, as long as it is the same as the gap between the nth and the n-1th. Most of said even numbers are multiples of three, as a result of the pigeonhole principle. If the difference is not a multiple of three, one of the three numbers n-x, n or n+x will be a multiple of three.
Examples
- 53 is a balanced prime. The previous prime number is 47, while the next prime number is 59, both of which are 6 numbers away from 53.
- 5 is a balanced prime. The pervious prime number is 3, while the next prime number is 7, both of which are 2 numbers away from 5.